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Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics-Informed Neural Networks(PINNs)
Ashutosh K. Mishra1 (PhD Student) Advisor: Emma Tolley
27 August 2025
1 1Email: ashutosh.mishra@epfl.ch
CMB Power Spectrum
BM:DM = 1:5 !
ΛCDM Theoretical Fit:
Credit: ESA and the Planck Collaboration
Ωbh2 ≈ 0.024, Ωmh2 ≈ 0.14
2
Small Scale Challenges in CDM Model
Potential Problem: Absence of Baryonic Processes (Feedback, Formation) and/or Nature of DM!
3
Oman et.al. 2015
Baryonic Processes
Strongly model dependent e.g. feedback sensitivity to the gas threshold for galaxy formation.
Very Difficult to disentangle baryonic effects in the Simulations!
Some outliers like IC 2574
still unexplainable with Feedback!
4
Alternative Dark Matter Models
Warm Dark Matter (WDM): favored mass range in tension with Ly observation & abundance of high-z galaxies
Self-interacting Dark Matter (SIDM): Needs fine-tuned cross-sections & struggles to explain full range of observations
Still a viable window
No Empirical
Strong constraints
arXiv:2408.00082
Under observational constraints!
Detection so far!
5
with microlensing observations!
Fuzzy Dark Matter
(F(C)DM, BECDM, ULDM, ELBDM, (ultra-light) axion (-like) DM (ULA, ALP))
Extremely light scalar particle (m 10-20 - 10-22 eV)
Non-thermally produced (thus not ultra-hot)
Clumps to form Bose-Einstein Condensate (BEC)!
Quantum effects counteract gravity at small scales
Tiny mass
large de-broglie wavelength ( 1/m)
macroscopic quantum effects at kpc scales
6
7
FDM potential signature in 21cm power spectrum
Distinct power suppression at high z!
8
FDM mass constraints based on PLin 
Need Full Non-
Linear Simulations to get reliable constraints!
9
Governing Equations
Wave Formalism (Schrödinger-Poisson Equations)
Mean Field Interpretation: Single Macroscopic WF of BEC
i ℏ2 2
ℏ∂tψ = − 2m ∇ ψ + mVψ
∇2V = 4πGm( | ψ |2 − | ψ0 |2 )
ρ
m
Madelung Formalism (Fluid Dynamics Representation)
∂tρ + ∇⃗ ⋅ (ρv⃗) = 0
1
ℏ2 ∇2
ψ = eiS
ρ
∂tv⃗ + (v⃗ ⋅
∇⃗)v⃗ = − m
∇⃗(V −
2m )
ρ = m | ψ |2
ρ
ℏ
= Q
∇2V = 4πGm(ρ − ρ0)
v = m ∇S
Q ill-defined at ρ = 0 !
“Quantum Pressure”
10
Fluid Solver unable to capture interference effects!
Fuzzy Dark Matter Simulations
Stick to SP-Equations for evolution!
11 A. Kunkel et.al. 2024
Challenges in Simulating Fuzzy Dark Matter
Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution
Time step scaling: Δt ∼ Δx2
Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation
For FDM evolution?)
So far sims. restricted to small box sizes of 10Mpc/h
12 Schive, Chieuh, & Broadhurst (2014)
Challenges in Simulating Fuzzy Dark Matter
Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution
Time step scaling: Δt ∼ Δx2
Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation
For FDM evolution?)
So far sims. restricted to small box sizes of 10Mpc/h
13 Schive, Chieuh, & Broadhurst (2014)
Challenges in Simulating Fuzzy Dark Matter
Both Mpc-scale and kpc-scales need to be Resolved for accurate evolution
Time step scaling: Δt ∼ Δx2
Hydrodynamical codes are used in N-body Simulation (but Fluid Formulation
For FDM evolution?)
So far sims. restricted to small box sizes of 10Mpc/h
14 Schive, Chieuh, & Broadhurst (2014)
Physics Informed Neural Networks
General Framework:
𝒟[NN(X, θ); λ] = f(X), X ∈ Ω
ℬ[NN(X, θ); ] = g(X) X ∈ ∂Ω
Custom Loss Function: with PDE and boundary conditions as additional constraints
Adapted from F. Pioch et.al.2023
Pretty Successful in Fluid and Climate Simulations!
Raissi, Yazdani, Karinadakis 2020
15
Outline of the approaches
01
Coordinate based approach
Directly solve the Schrödinger–Poisson (SP) equations using Physics-Informed Neural Networks (PINNs).
02
Generative Modeling
Learn the mapping from Cold Dark Matter (CDM) to Fuzzy Dark Matter (FDM) cosmological simulations at a fixed redshift.
03
Physics-Informed Generative Modeling
Interpolate and simulate FDM cosmological boxes across multiple redshifts using physics-constrained generative models.
16
Schrödinger-Poisson Equations used
λ = ℏ ⟹
1
potential
λ : the strength of
m i ∂ Ψ(x, t) = − λ ∇2 + 1 V[Ψ(x, t)] Ψ(x, t)
∂t ( 2 λ )
∇2V[Ψ(x, t)] = ( |Ψ(x, t) |2 − 1)
λ → 0, Gravitational Potential Term is dominant in the SP Equations!
λ → ∞, Gravitational Potential Term vanishes, Free Schrodinger Equation
representing diffusion!
λ = 1 throughout this work!
17
x
y
Re(ψ)
Schrödinger-Poisson Informed Neural Networks (SPINN)
iℏ∂ ψ = − ℏ
2
t
2m
∇2ψ + mVψ
∇2V = 4πGm( | ψ |2 − | ψ0 |2 )
z
t
Neural network
Im(ψ)
V
18
x
y
Re(ψ)
Schrödinger-Poisson Informed Neural Networks (SPINN)
iℏ∂ ψ = − ℏ
2
t
2m
∇2ψ + mVψ
∇2V = 4πGm( | ψ |2 − | ψ0 |2 )
z
t
Neural network
Im(ψ)
V
Use automatic differentiation to calculate derivatives
∂tψ, ∇2ψ, ∇2V, . . .
19
x
y
Re(ψ)
Use automatic differentiation to calculate derivatives
∂tψ, ∇2ψ, ∇2V, . . .
Schrödinger-Poisson Informed Neural Networks (SPINN)
iℏ∂ ψ = − ℏ
2
t
2m
∇2ψ + mVψ
∇2V = 4πGm( | ψ |2 − | ψ0 |2 )
Construct physics-informed loss:
L = | (i∂ψ/∂t + ∇2ψ/2 − mVψ) |2 + | ( ∇2V − | ψ2 | + 1) |2
z
t
Neural network
Im(ψ)
V
20
x
y
Re(ψ)
Use automatic differentiation to calculate derivatives
∂tψ, ∇2ψ, ∇2V, . . .
Use backpropagation to find network parameters that minimize these losses
Schrödinger-Poisson Informed Neural Networks (SPINN)
Loss for initial and boundary conditions
Construct physics-informed loss:
L = | (i∂ψ/∂t + ∇2ψ/2 − mVψ) |2 + | ( ∇2V − | ψ2 | + 1) |2
z
t
Neural network
Im(ψ)
V
21
Results
Density Predictions
1D NN
1D Sim
Mishra & Tolley, ApJ 2025
3D NN
3D Sim
Unsupervised neural network accurately predicting FDM dynamics using just physics constraints and initial conditions
23
Extrapolation
Evidence of SPINNs' ability to generalize and predict beyond trained time intervals !
24
What about cosmological FDM simulations?
Approaches
Learning the CDM to FDM density mapping
ρCDM(x) → ρFDM(x)
(Essence: Learn the small scale corrections with ML — where FDM diverges — while taking advantage of its numerical equivalence to CDM at large scales.)
Use physics-informed Generative models to simulate/interpolate FDM boxes across redshifts
25
How do diffusion models work?
source: arXiv:1503.03585, 2015
Source: medium.com – Vadim Titko 26
Data for CDM to FDM Mapping I
Courtesy of Simon May
We start with 10 Mpc simulation boxes for both Cold Dark Matter (CDM) and Fuzzy Dark Matter (FDM), covering the same region in space.
Original FDM, CDM box resolution: 86403, 20483
27
Data for CDM to FDM Mapping II
For both the boxes, we extract a sub-volume spanning 0 to 3 Mpc/h on each side.
This 3 Mpc region is then:
 |
Further chunked into smaller 64³ voxel boxes to fit within GPU memory constraints.
In total 729 pairs of CDM and FDM boxes!
28
Current Hybrid Approach-WGAN (Ongoing work)
Real (FDM)
Generative Model (WGAN)
Input (CDM)
Generated (FDM)
29
Fringes not fully resolved!
Current Hybrid Approach-WGANs (Ongoing work)
Real
Real (FDM)
30
Generated (FDM)
Much better!
Current Hybrid Approach-DDPMs(Ongoing work)
Real
Real (FDM)
31
Can be used as an effective compression algorithm for FDM
Training FDM Data size 130 GB, CDM Data 0.8 GB, DDPM Model 1.3 GB
Input (CDM)
Generative Model (DDPM)
Generated (FDM)
130GB
Compressed to
2GB
(FDM) (CDM + DDPM Model)
65 fold compression!
32
What about generalizing to unseen regions? Can these models hallucinate realistically?
Testing the model trained on 0-3 Mpc sided volume beyond the trained spatial region
Actual: 3-6 Mpc volume
DDPM @ test
33
What about generalizing to unseen regions? Can these models hallucinate realistically?
When trained with larger volume cut-outs and some additional losses!
Overlapping patches of 1.5 Mpc
Overlapping patches of 1.5 Mpc
+ Powerspectrum loss
34
Can these models hallucinate realistically?
May be with
Testing the model trained on 0-3 Mpc sided volume beyond the trained spatial region
Actual: 3-6 Mpc volume
DDPM @ test
35
Data for Physics-Informed GANs
We have 2 Mpc/h sized FDM boxes at redshifts z = {127, 63, 31, 15, 7, 6, 5, 4, 3, 2, 1, 0}.
Out of which z = {7, 6, 5, 4, 3, 2} boxes are chosen for training and z = {1, 0} for test.
Each of these boxes’ original resolution is
24003
This chosen subset of samples is then:
|
36
(Physics-Informed) Generative Models (in progress)
After 8 hrs of training on single GPU
Real
Gen
Yet to include full physics!
∇2Y |pred = ∇2Y |actual
37
Neural Operators for CDM to FDM mapping
Physics-informed neural operators for painting in FDM small-scale
features
Supervised PINNs using large-scale CDM simulations as additional data
constraint
Reproducing Core-Halo Relations for FDM with PINNs
38
Question?
Arxiv Link
39
Back-up
Generative Adversarial Networks (GANs)
Real Images
High Dimensional Sample Space
Generator
G
Discriminator
D
Generated Images
Real
Low Dimensional Latent Space
Fake
41
Fuzzy Dark Matter
Linear theory predicts sharp cutoff in power spectrum due to quantum pressure
42
Direct probe of DM distribution
Power spectrum, halo mass profile
Bechtol et al. 2023
43
Numerical Method (Mocz et. al. 2017)
2nd Order Unitary Spectral Method
Calculate potential:
V = IFFT − 1 FFT 4πGm( | ψ |2 − | ψ |2 )
Half-Step ‘Kick’:
( k2 (
0 ))
ψ ← exp[−i(m/ℏ)(Δt/2)V ]ψ
Full-Step ‘Drift’ in Fourier Space:
ψ ← IFFT (exp[−iΔt(ℏ/m)k2 /2]FFT(ψ))
Kick
Drift
Update the potential:
V ← IFFT
1 FFT
4πGm( | ψ |2 − | ψ |2 )
Another Half-Step ‘Kick’:
( k2 (
0 ))
ψ ← exp[−i(m/ℏ)(Δt/2)V ]ψ
Kick